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On the mixed derivatives of a separately twice differentiable function. (English) Zbl 1393.26011
The main result of the paper is the following
Theorem. Let $$f:[0,1]^2\rightarrow\mathbb{R}$$ be a separately twice differentiable function and $$f_{xx}^{''},f_{yy}^{''}\in L^2([0,1]^2)$$. Then:
1) mixed derivatives $$f_{xy}^{''}$$ and $$f_{yx}^{''}$$, which are equal and $$f_{xy}^{''},f_{yx}^{''}\in L^2([0,1]^2)$$, exist almost everywhere;
2) a set $$A\subset [0,1]$$ with $$\mu(A)=1$$ exists such that $$f_{x}^{'}$$ is continuous with respect to $$y$$ at every point of the set $$A\times [0,1]$$;
3) $$f\in C([0,1]^2)$$.
Note that the condition of square integrability of $$f_{xx}^{''}$$ and $$f_{yy}^{''}$$ is essential, namely, in the [V. Mykhaylyuk and A. Plichko, Colloq. Math. 141, No. 2, 175–181 (2015; Zbl 1339.26029)] the authors constructed a separately twice differentiable function $$f:[0,1]^2\rightarrow\mathbb{R}$$ for which the mixed partial derivative $$f_{xy}^{''}$$ does not exist on a set of positive measure.

##### MSC:
 26B05 Continuity and differentiation questions
##### Keywords:
mixed derivatives; Mazur problem; continuity; Fourier series
Zbl 1339.26029
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